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Unit and Proper Tube Orders

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Abstract

In 2005, we defined the n-tube orders, which are the n-dimensional analogue of interval orders in 1 dimension, and trapezoid orders in 2 dimensions. In this paper we consider two variations of n-tube orders: unit n-tube orders and proper n-tube orders. It has been proven that the classes of unit and proper interval orders are equal, and the classes of unit and proper trapezoid orders are not. We prove that the classes of unit and proper n-tube orders are not equal for all n ≥ 3, so the general case follows the situation in 2 dimensions.

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References

  1. Bogart, K.P., West, D.B.: A short proof that ‘proper=unit’. Discrete Math. 201, 21–23 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bogart, K.P., Möhring, R.H., Ryan, S.P.: Proper and unit trapezoid orders and graphs. Order 15, 325–340 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  3. Dagan, I., Golumbic, M.C., Pinter, R.Y.: Trapezoid graphs and their coloring. Discrete Appl. Math. 21(1), 35–46 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  4. Felsner, S., Müller, R., Wernisch, L.: Trapezoid graphs and generalizations, geometry and algorithms. Discrete Appl. Math. 74, 13–32 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  5. Fishburn, P.C.: Intransitive indifference with unequal indifference intervals. J. Math. Psychol. 7, 144–149 (1970)

    Article  MATH  MathSciNet  Google Scholar 

  6. Fishburn, P.C.: Interval Orders and Interval Graphs: A Study of Partially Ordered Sets. AT&T Bell Laboratories, Bridgewater (1985)

    MATH  Google Scholar 

  7. Fishburn, P.C.: Generalizations of semiorders: a review note. J. Math. Psychol. 41(4), 357–366 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  8. Laison, J.D.: Tube representations of ordered sets. Order 21(3), 209–232 (2005)

    MathSciNet  Google Scholar 

  9. Roberts, F.S.: Indifference graphs. In: Harary, F. (ed.) Proof Techniques in Graph Theory, pp. 139–146. Academic, London (1969)

    Google Scholar 

  10. Ryan, S.P.: Trapezoid order classification. Order 15, 341–354 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  11. Ryan, S.P.: Trapezoid orders: geometric classification and dimension. PhD thesis, Dartmouth College, Hanover (1999)

  12. Trotter, W.T.: Combinatorics and Partially Ordered Sets: Dimension Theory. The Johns Hopkins University Press, Baltimore (1992)

    MATH  Google Scholar 

  13. Trotter, W.T.: New perspectives on interval orders and interval graphs. In: Bailey, R.A. (ed.) Surveys in Combinatorics 1997, pp. 237–286. Cambridge University Press, Cambridge (1997)

    Google Scholar 

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  1. Mathematics Department, Willamette University, 900 State Street, Salem, OR, USA

    Joshua D. Laison

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  1. Joshua D. Laison
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Correspondence to Joshua D. Laison.

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Laison, J.D. Unit and Proper Tube Orders. Order 25, 237–242 (2008). https://doi.org/10.1007/s11083-008-9091-7

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  • DOI: https://doi.org/10.1007/s11083-008-9091-7

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